Taylor Series Handout - 21-122 Integration and Approximation TA Bryan Ding([email protected] April 4 2019 Introduction to Taylor Series Practice

# Taylor Series Handout - 21-122 Integration and...

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21-122: Integration and Approximation TA: Bryan Ding ([email protected]) April 4, 2019 Introduction to Taylor Series Practice with Taylor Series Theorem Iffhas a power series representation (expansion) ata, that is, iff(x) =Xn=0cn(x-a)n,where|x-a|< R,then its coefficients are given by the formulacn=f(n)(a)n!.Taylor SeriesThe Taylor Series Expansion for a functionf(x) centered aroundx=ais given by:f(x) =Xn=0f(n)(a)n!(x-a)n. Practice Problems: 1. Find the Taylor series for cosh(x) centered atx= 0.Hint: Recall thatddx(cosh(x)) = sinh(x)andddx(sinh(x)) = cosh(x), along with the fact thatsinh(0) = 0andcosh(0) = 1. 1
2. Find the Taylor series for sinh(x) centered atx= 0 using problem 1.3. Find the Taylor series for sinh(x) centered atx= 0 using sinh(x) =ex-e-x2.4. The Taylor series forexconverges for allx, even whenxis a complex number. Using this fact, constructthe Taylor series foreto prove Euler’s formulae= cosθ+isinθ. 2